I am so sorry about your kid. Usually you will see the head with the feet there. Its possible a leg was bent back or something. It sounds like you did all you could in the circumstances. It is very easy to read about things in books, but when they go wrong in real life, its a different matter. If you have livestock, these things happen.
I have had goats etc. many years. This year, we have had two dead billy kids at birth. I was monitoring both goats regularly, but they kidded in between visits. Had I been there, I may, or may not have been able to do anything. But, its no use beating yourself up, your nanny goat is still with you. She will be able to have more kids and next time things will go well. I was fortunate that another of my goats had twin females a few weeks later, and this cheered me up no end.
I did not milk the goat who lost the kid as it was a pygmy, and the other had a surviving kid. If you want to have goats milk, they you can start milking her. Although don't use it for the first day or two, as it will be strong and yellow, but ok for cats!! Just remember though, once you start to milk her, you will need to carry on.